

Nanosystems
The most talked about technical book of the year.
-Bob Schwabach
United Press International
With this book, Drexler has established the field of molecular nanotechnology. The detailed analyses show quantum chemists and synthetic chemists how to build upon their knowledge of bonds and molecules to develop the manufacturing systems of nanotechnology, and show physicists and engineers how to scale down their concepts of macroscopic systems to the level of molecules.
-William A. Goddard III
Professor of Chemistry and Applied Physics
Director, Materials and
Molecular Simulation Center
California Institute of Technology
Devices enormously smaller than before will remodel engineering, chemistry, medicine, and computer technology. How can we understand machines that are so small? Nanosystems covers it all: power and strength, friction and wear, thermal noise and quantum uncertainty. This is the book for starting the next century of engineering.
-Marvin Minsky
Professor of Electrical Engineering
and Computer Science
Toshiba Professor of Media Arts and Sciences
Massachusetts Institute of Technology
What the computer revolution did for manipulating data, the nanotechnology revolution will do for manipulating matter, juggling atoms like bits... This multidisciplinary synthesis opens the door to the new field of molecular manufacturing.
-Ralph Merkle
Member of the Research Staff
Computational Nanotechnology Project
Xerox Palo Alto Research Center
This work provides the scientific and technological foundation for the emerging field of molecular systems engineering...It is essential for anyone contemplating research in this area...a milestone in the development of the technologies that will underpin the final industrial revolution.
-John Walker
Cofounder of Autodesk, Inc.
It is a scholarly examination of how this technology works, a reference book for the crafters of the future.
Demonstrates not only that nanotechnology is achievable, but shows how it will happen....takes readers from the fundamental physical principles to advanced designs for molecular components and systems.
\text { _Japan Times }Winner, 1992 Award for Best Computer Science Book.
-Association of American Publishers
Nanosystems
Molecular Machinery, Manufacturing, and Computation
K. Eric Drexler
Research Fellow
Institute for Molecular Manufacturing
Palo Alto, California
A WILEY-INTERSCIENCE PUBLICATION
John Wiley & Sons, Inc.
NEW YORK - CHICHESTER - BRISBANE - TORONTO - SINGAPORE
Excerpt from "There's Plenty of Room at the Bottom" by Richard P. Feynman (C) 1960 by Engineering & Science, California Institute of Technology; used with permission.
In recognition of the importance of preserving what has been written, it is a policy of John Wiley & Sons, Inc. to have books of enduring value published in the United States printed on acid-free paper, and we exert our best efforts to that end.
Copyright (C) 1992 by John Wiley & Sons, Inc.
All rights reserved. Published simultaneously in Canada.
Reproduction or translation of any part of this work beyond that permitted by section 107 or 108 of the 1976 United States Copyright Act without the permission of the copyright owner is unlawful. Requests for permission or further information should be addressed to the Permission Department, John Wiley & Sons, Inc.
Library of Congress Cataloging-in-Publication Data
Drexler, K. Eric.
Nanosystems : molecular machinery, manufacturing, and computation / K. Eric Drexler. p. .
"A Wiley-Interscience publication."
Includes bibliographical references and index.
ISBN 0-471-57547-X. -- ISBN 0-471-57518-6 (pbk.)
- Nanotechnology. I. Title.
T174.7.D74 1992
620.4--dc20
CIP
To experimentalists, engineers, and software builders:
they do the hard parts
Preface
Manufactured products are made from atoms, and their properties depend on how those atoms are arranged. This volume summarizes 15 years of research in molecular manufacturing, the use of nanoscale mechanical systems to guide the placement of reactive molecules, building complex structures with atom-byatom control. This degree of control is a natural goal for technology: Microtechnology strives to build smaller devices; materials science strives to make more useful solids; chemistry strives to synthesize more complex molecules; manufacturing strives to make better products. Each of these fields requires precise, molecular control of complex structures to reach its natural limit, a goal that has been termed molecular nanotechnology.
It has become clear that this degree of control can be achieved. The present volume assembles the conceptual and analytical tools needed to understand molecular machinery and manufacturing, presents an analysis of their core capabilities, and explores how present laboratory techniques can be extended, stage by stage, to implement molecular manufacturing systems. It says little about applications other than computation (describing -instruction-per-second submicron scale CPUs executing instructions per second per watt) and manufacturing (describing desktop devices able to produce precisely structured, kilogram-scale products from simple chemical feedstocks). Surveys of broader implications appear elsewhere (Drexler, 1986a; 1989c; Drexler et al., 1991).
The intended readership
Molecular manufacturing is linked to many areas of science and technology. In writing this volume, I have been guided by an imaginary committee of readers with differing demands.
One is a reader with a general science background, interested in the basic principles, capabilities, and nature of molecular nanotechnology, but not in the mathematical derivations. Accordingly, I have attempted to summarize the chief results in descriptions, diagrams, and example calculations, and have included comparisons of this field to others that are more familiar. Such a reader can skip many sections without becoming lost.
Another is a student considering a career in the field. This reader demands an introduction to the foundations of molecular nanotechnology presented in terms of the basic physics, calculus, and chemistry taught to students in other fields. Accordingly, I have grounded most derivations in basic principles, developing intermediate results as needed.
The rest of the committee includes a physicist, a chemist, a molecular biologist, a materials scientist, a mechanical engineer, and a computer scientist. Each has deep professional knowledge of a particular field. Each demands answers to special questions that presuppose specialized knowledge. Each knows the exceptions that hide behind most generalizations, and the approximations that hide behind most textbook formulas. Accordingly, the discussion sometimes dives into a topic that readers outside the relevant discipline may find opaque. Skipping past these topics will seldom impair comprehension of what follows.
Each of these specialists also represents a community of researchers able to advance the development of molecular nanotechnology. Accordingly, many of the discussions implicitly or explicitly highlight open problems, inviting work in theoretical analysis, in computer-aided design and modeling, and in laboratory experimentation. I hope that this volume will be seen both as a guide and as an invitation to a promising new field.
The nature of the subject
Our ability to model molecular machines-of specific kinds, designed in part for ease of modeling - has far outrun our ability to make them. Design calculations and computational experiments enable the theoretical study of these devices, independent of the technologies needed to implement them. Work in this field is thus (for now) a branch of theoretical applied science (Appendix A).
Molecular manufacturing applies the principles of mechanical engineering to chemistry (or should one say the principles of chemistry to mechanical engineering?) and uses results drawn from materials science, computer science, and elsewhere. But interdisciplinary studies can foster misunderstandings. From every disciplinary perspective, a superficial glance suggests that something is wrongapplying chemical principles leads to odd-looking machines, applying mechanical principles leads to odd-looking chemistry, and so forth. The following chapters offer a deeper view of how these principles interact.
Criticism of criticism
Research in molecular nanotechnology requires a design perspective because it aims to describe workable systems. It is easy to describe unworkable systems, and criticisms of a critic's own bad design have on occasion been presented as if they were criticisms of molecular nanotechnology as a whole. Some examples: assuming the use of flexible molecules, then warning that they will have no stable shape; assuming the manipulation of unbound reactive atoms, then warning that they will react and bond to the manipulator; assuming the use of materials with unstable surfaces, then warning that the surfaces will change; assuming that reactive gases permeate nanosystems, then warning that reactions will occur; assuming that nanomachines must "see," then warning that light waves are too long and -rays too energetic; assuming that nanomachines swim from point to point, then warning that Brownian motion makes such navigation impossible; assuming that nanomachines dissipate enormous power in a small volume, then warning of overheating; and so on, and so forth. These observations constitute not criticisms, but rediscoveries of elementary engineering constraints.
Use of tenses
In ordinary discourse, "will be" suggests a prediction, while "would be" suggests a conditional prediction. Using these future-tense expressions is inappropriate when discussing the time-independent possibilities inherent in physical law.
In speaking of spacecraft trajectories to Pluto, for example, to say that they "will be" is to predict the future of spaceflight; to say that they "would be" is to remind readers of the uncertainties of budgets and life. Both phrases distract from the analysis of celestial mechanics and engineering trade-offs. The present tense is more serviceable: One can say that as-yet unrealized spacecraft trajectories to Pluto "are of two kinds, direct and gravity assisted," and then analyze their properties without distraction. Similarly, one can say that as-yet unrealized nanomachines of diamondoid structure "are typically stiffer and more stable than folded proteins." Much of the discussion in this volume is cast in this timeless present tense; this is not intended to imply that devices like those discussed in Parts I and II presently exist.
Citations and apologies
It is much easier to grasp and apply the main results of a field than it is to provide a balanced guide to the recent work, omitting no useful citations. I am sure that my discussions of chemistry and protein engineering, for example, omit papers fully as valuable as the best included. I apologize to authors I have slighted.
Less forgivable are those instances (which I cannot yet identify) in which I may have rederived some result that should be attributed to an earlier author, perhaps well known in some specialty. In interdisciplinary research, one cannot spend a professional lifetime immersed in a single literature, and such failures of attribution become likely-mathematics often yields results more easily than does a library. Any such lapses brought to my attention will be corrected in future editions; their most likely locations are Chapters 5, 6, and 7.
Aside from these lapses, material presented without citation falls into two categories that I trust are distinct. First, well-known principles and results from established fields-physics, statistical mechanics, chemistry-are used without citing Newton, Boltzmann, Pauling, or their kin. Second, the designs, concepts, and analytical results that are both specific to nanomechanical systems and not attributed to someone else are to the best of my knowledge original contributions, many presented for the first time in this volume.
It also seems necessary to apologize for doing theoretical work in a world where experimental gains are often so hard-won. If this theoretician's description of possibilities seems to make light of experimental difficulties, I can only plead that it would soon become tedious to say, at every turn, that laboratory work is difficult, and that the hard work is yet to be done.
Acknowledgments
The research behind this volume began in 1977 , stimulated by the growing literature on biological molecular machines. Basic results appeared in a refereed paper (Drexler, 1981). The present work began as notes for a course taught at Stanford in 1988 at the invitation of Nils Nilsson; early versions of some chapters did service as a doctoral thesis at MIT in 1991. During this long gestation, many people have contributed through discussion, criticism, and detailed review.
I thank the participants of the monthly series of nanotechnology seminars (some centered on draft chapters of this volume) organized by Ralph Merkle at the Xerox Palo Alto Research Center for wide-ranging discussion and criticism. These have included Lakshmikantan Balasubramaniam, David Biegelsen, Ross Bringans, David Fork, Babur Hadimioglu, Stig Hagstrom, Conyers Herring, Tad Hogg, Warren Jackson, Noble Johnson, Martin Lim, Jim Mikkelsen, John Northrup, K. V. Ravi, Paolo Santos, Mathias Schnabel, Bob Street, Lars-Erik Swartz, Eugen Tarnower, Dean Taylor, Rob Tow, and Chris Van der Walle.
For reviews, suggestions, and specific pieces of help, I thank Jeff Bottaro, Randall Burns, Jamie Dinkelacker, Greg Fahy, Jonathan Goodman, Josh Hall, Robin Hanson, Norm Hardy, Ted Kaehler, Markus Krummenacker, Arel Lucas, Tim May, John McCarthy, Mark Samuel Miller, Chip Morningstar, Russell Parker, Marc Stiegler, Eric Dean Tribble, John Walker, and Leonard Zubkoff.
For discussion and suggestions that helped in preparing a 1989 draft paper that became Section 15.3, I thank Joe Bonaventura, Jeff Bottaro, William DeGrado, Bruce Erickson, Barbara Imperiali, Jim Lewis, Danute Nitecki, Chris Peterson, Fredric Richards, Jane Richardson, and Kevin Ulmer. For similar help in developing ideas in Section 15.4, I thank Tom Albrecht, John Foster, Paul Hansma, Jan Hoh, Ted Kaehler, Ralph Merkle, Klaus Mosbach, and Craig Prater. For sponsoring the initial 1981 publication, I thank Arthur Kantrowitz.
For remarkable efforts while this work was on its way to fulfilling the thesis requirement of an interdepartmental doctoral program hosted by the Media Arts and Sciences Section at MIT, I thank the committee's chair, Marvin Minsky (Department of Electrical Engineering and Computer Science; Media Arts and Sciences Section), as well as committee members Alexander Rich (Department of Biology), Gerald Sussman (Department of Electrical Engineering and Computer Science), Rick Danheiser (Department of Chemistry), and Steven Kim (Department of Mechanical Engineering), with special thanks to Steve Benton and Nicholas Negroponte (Media Arts and Sciences Section) for making the program possible in an environment hospitable to new research directions.
Ralph Merkle has helped greatly by providing steady encouragement and extensive opportunities for discussion during the writing of this volume, by reviewing it (and helping to obtain other reviews), and by collaborating on several of the design studies described. Special thanks also go to Jeffrey Soreff, whose checking of mathematical results and physical reasoning has been just barely incomplete enough for him to escape blame for the remaining errors: with his other help, this places his contribution in a class by itself. Barry Silverstein, John Walker, and the Institute for Molecular Manufacturing each provided essential support for a major portion of this work. Diane Cerra and Bob Ipsen of John Wiley & Sons made publication a pleasure. Chris Peterson, my spouse and partner, provided essential support of kinds too numerous to list.
At the other pole of involvement, the most general thanks go to members of the hundred or more audiences at universities and industrial laboratories in the U.S., Europe, and Japan who have listened to presentations of these ideas and aided in their development by intelligent questioning. I hope that this volume provides many of the answers they sought.
Introduction and Overview
1.1. Why molecular manufacturing?
The following devices and capabilities appear to be both physically possible and practically realizable:
- Programmable positioning of reactive molecules with precision
- Mechanosynthesis at operations/device second
- Mechanosynthetic assembly of objects in
- Nanomechanical systems operating at
- Logic gates that occupy
- Logic gates that switch in and dissipate
- Computers that perform instructions per second per watt
- Cooling of cubic-centimeter, systems at
- Compact MIPS parallel computing systems
- Mechanochemical power conversion at
- Electromechanical power conversion at
- Macroscopic components with tensile strengths
- Production systems that can double capital stocks in
Of these capabilities, several are qualitatively novel and others improve on present engineering practice by one or more orders of magnitude. Each is an aspect or a consequence of molecular manufacturing.
1.2. What is molecular manufacturing?
This volume describes the fundamental principles of molecular machinery and applies them to nanomechanical devices and systems, including molecular manufacturing systems and computers. At present, however, these are unfamiliar topics. New fields often need new terms to describe their characteristic features, and so it may be excusable to begin with a few definitions: Molecular manufacturing is the construction of objects to complex, atomic specifications using sequences of chemical reactions directed by nonbiological molecular machinery. Molecular nanotechnology comprises molecular manufacturing together with its techniques, its products, and their design and analysis; it describes the field as a whole. Mechanosynthesis-mechanically guided chemical synthesis-is fundamental to molecular manufacturing: it guides chemical reactions on an atomic scale by means other than the local 'steric* and electronic properties of the reagents; it is thus distinct from (for example) enzymatic processes and present techniques for organic synthesis.
At the time of this writing, positional chemical synthesis is at the threshold of realization: precise placement of atoms and molecules has been demonstrated (for example, see Eigler and Schweizer, 1990), but flexible, extensible techniques remain in the domain of design and theoretical study (Part III), as does the longer-term goal of molecular manufacturing (Chapter 14). Accordingly, the implementation of molecular nanotechnologies like those analyzed in Part II awaits the development of new tools. This volume is addressed both to those concerned with identifying promising directions for current research, and to those concerned with understanding and preparing for future technologies.
The following chapters form three parts: Part I describes the chief physical principles and phenomena of importance in molecular machinery and manufacturing. Part II applies the results of Part I to the design and analysis of components and systems (yielding the conclusions summarized in Section 1.1). Part III then describes implementation pathways leading from our current technology base to systems like those described in Part II.
The rest of the present section attempts to clarify the nature of the topic by discussing an example of a nanomechanical device and by presenting a chemical perspective on molecular manufacturing. Sections 1.3 to 1.5 present a set of comparisons between this and other fields (mechanical engineering, microtechnology, chemistry, and molecular biology), a discussion of overall approach (including objectives, level, scope, and assumptions), and an overview of the later chapters and how they fit together. Table 1.1 lists some of the known problems and constraints that are addressed elsewhere in this volume.
1.2.1. Example: a nanomechanical bearing
As discussed in Section 1.3, the mechanical branch of molecular nanotechnology forms a field related to, yet distinct from, mechanical engineering, microtechnology, chemistry, and molecular biology. An example may serve as a better introduction than would an attempt at a written definition.
Figure 1.1 shows several views of one design for a nanomechanical bearing like those discussed in greater depth in Chapter 10. (Figure 1.2 describes some conventions used in illustrations like Figure 1.1) In a functional context, many of the bonds shown as hydrogen terminated would instead link to other moving parts or to a structural matrix. Several characteristics are worthy of note:
- The components are polycyclic, more nearly resembling the fused-ring structures of diamond than the open-chain structures of biomolecules such as proteins.1
Table 1.1. Some known issues, problems, and constraints.
Thermal excitation Thermal and quantal positional uncertainty Quantum-mechanical tunneling Bond energies, strengths, and stiffnesses Feasible chemical transformations Electric field effects Contact electrification Ionizing radiation damage Photochemical damage Thermomechanical damage Stray reactive molecules Device operational reliabilities Device operational lifetimes Energy dissipation mechanisms Inaccuracies in molecular mechanics models Limited scope of molecular mechanics models Limited scale of accurate quantal calculations Inaccuracy of semiempirical models Providing ample safety margins for modeling errors
- Accordingly, each component is relatively stiff, lacking the numerous opportunities for internal rotation about bonds that make conformational analysis difficult in typical biomolecules.
- Repulsive, nonbonded interactions strongly resist both rotations of the shaft away from its axial alignment with the ring, and displacement along that axis or perpendicular to it.
- Rotation of the shaft about its axis within the ring encounters negligible energy barriers, indicating a nearly complete absence of static friction.
- The combination of stiffness in five degrees of freedom with facile rotation in the sixth makes the structure act as a good bearing, in the conventional mechanical engineering sense of the term.
- The absence of significant static friction in a system that places bumpy surfaces in firm contact with no intervening lubricant is surprising by conventional mechanical engineering standards.
- No solvent is illustrated, and there is no reason to think that the bearing structure shown would in fact be soluble.
- Neither of the components of the bearing is a plausible target for synthesis using reagents diffusing in solution; barring unprecedented chemical cleverness, their construction requires mechanosynthetic control.
How typical are these characteristics? Stiff, polycyclic structures are ubiquitous in the designs presented in Part II. Many components are designed to combine stiff constraints in some degrees of freedom with nearly free motion in others, thereby fulfilling roles familiar in mechanical engineering; nonetheless,

side view
Figure 1.1. End views and exploded views of a sample overlap-repulsion bearing design (shown in both ball-and-stick and space-filling representations). Geometries represent energy minima determined by the MM2/CSC molecular mechanics software. Note the six-fold symmetry of the shaft and fourteen-fold symmetry of the surrounding ring; with a least common multiple of 42 , this combination yields low energy barriers to rotation of the shaft within the ring. Bearing structures are discussed further in Chapter 10. (MM2/CSC denotes the Chem3D Plus implementation of the MM2 molecular mechanics force field. The MM2 model is discussed in Section 3.3.2; Chem3D Plus is a product of Cambridge Scientific Computing, Cambridge, Massachusetts.)

Figure 1.2. Conventions for atom representation using shading and relative sizes. The "H (fixed)" atom represents a hydrogen atom held at fixed spatial coordinates; these are used to model some mechanical constraints applied by a larger structure (e.g., in stiffness calculations). All radii are set equal to the values for compressive contacts given by Eq. (3.20).
a detailed understanding of how those roles are fulfilled requires analyses based on uniquely molecular phenomena. The operating environment assumed for the nanomechanical and mechanosynthetic systems discussed in Part II is high vacuum, rather than a solvent. Finally, the designs in Part II (unlike those described in Part III) are consistently of a scale and complexity that precludes synthesis using present techniques.
The bearing shown in Figure 1.1 suggests the nature of other systems described in Part II. For example, the combination of a bearing and shaft suggests the possibility of extended systems of power-driven machinery. The outer surface of the bearing suggests the possibility of a molecular-scale gear. The controlled rotary motion of the shaft within the ring, together with the concept of extended systems of machinery, suggests the possibility of controlled molecular transport and positioning, which is necessary for advanced mechanosynthesis.
1.2.2. A chemical perspective on molecular manufacturing
Chemistry today (and chemical synthesis in particular) focuses chiefly on the behavior of molecules diffusing and colliding in solution. Reaction rates in solution-phase chemistry are determined by multiple influences, including concentration-dependent collision frequencies, and steric and electronic effects local to the reacting molecules.
Although based on the same principles of physics, mechanosynthesis performed by molecular machinery in vacuum differs greatly from conventional chemistry. Concepts developed to describe diffusing molecules in a gas or liquid (or immobile molecules in a solid) often must be modified in describing systems characterized by nondiffusive mobility. The concept of "concentration," for example, in the familiar sense of "number of molecules of a particular type per unit of macroscopic volume" becomes inapplicable to calculations of reaction rates. Local steric and electronic effects remain important, but the decisive influence on reaction rates becomes mechanical positioning aided by applied force. a. Machine-phase systems. To emphasize differences from solid-, liquid-, and gas-phase systems, it can be useful to speak of machine-phase systems and chemistry:
- A machine-phase system is one in which all atoms follow controlled trajectories (within a range determined in part by thermal excitation).
- Machine-phase chemistry describes the chemical behavior of machine-phase systems, in which all potentially reactive moieties follow controlled trajectories.
The useful distinction between liquid phase and gas is blurred by the existence of supercritical fluids; the useful distinction between solid and liquid is blurred by the existence of glasses, liquid crystals, and gels. Where machinephase chemistry is concerned, the definitional ambiguities are chiefly associated with the words all and controlled. In a conventional chemical reaction or an enzymatic active site, a moderate number of atoms in a small region can be said to follow somewhat-controlled trajectories, but these examples fall outside the intended bounds of the definition. In a good example of a machine-phase system, large numbers of atoms follow paths that seldom deviate from a nominal trajectory by more than an atomic diameter while executing complex motions in an extended region from which freely-diffusing molecules are rigorously excluded. Machine-phase conditions can be termed "eutactic ("well arranged," from the Greek eus, "good," and taktikos, "of order or arrangement"). Eutactic conditions are quite unlike those of solution-phase chemistry.*
Mechanosynthesis of the sort discussed in Chapters 8 and 13 is a machinephase process (Part III discusses mechanosynthesis in a solvent environment). Eutactic mechanosynthesis offers novel chemical capabilities, such as positionbased discrimination among chemically equivalent sites, strong suppression of side reactions, and new sources of activation energy.
Chemistry in the machine phase shares characteristics of gas-, solution-, and solid-phase chemistry, and yet displays unique characteristics; these similarities and differences are discussed further in Sections 6.4.2 and 8.3. Since experience shows that habits of thought developed in the study of liquid- and gas-phase systems can yield misleading conclusions if hastily applied to machine-phase systems, frequent recourse to fundamental principles is necessary. The Index of this volume includes an entry (Chemistry: contrasts between machine and solution phase) that cites discussions of this issue. Table 1.2 provides a compact summary.
1.2.3. Exposition vs. implementation sequence
The implementation sequence for molecular manufacturing might proceed as follows: The ability to make complex molecular objects in solution is extended to objects of greater size and complexity. These molecular objects are used as components in molecular machines capable of directing the mechanosynthesis of yet larger and more complex machines. Through a series of steps, solution-2
Table 1.2. Contrasts between solution-phase and mechanosynthetic chemistry (a more detailed comparison appears in Table 8.1).
| Characteristic | Solution-phase chemistry | Mechanosynthetic chemistry |
|---|---|---|
| Reagent transport | Diffusion | Mechanical conveyance |
| Reaction site selectivity | Structural influences | Direct positional control |
| Reaction environment | Solution | Mechanisms in vacuum |
| Control of reaction environment | Relatively little | Relatively great |
| Intermolecular reactions | Ubiquitous opportunities | Strictly controlled opportunities |
| Unwanted reactions | Inter- and intramolecular | Chiefly intramolecular |
| Sensitivity to energy differences | is large | is (often) small |
| Typical product size | 10 to 100 atoms | atoms |
based mechanosynthetic methods are replaced by methods that require an inert environment, then polymeric building materials are replaced by diamondoid materials. Further increases in scale and capability yield advanced molecular manufacturing systems under computer control. (Each of these steps is discussed in Chapter 16.)
The expository sequence of this volume is quite different. It begins by describing fundamental principles of broad applicability (Part I), then applies them to the design and analysis of advanced systems (Part II). Finally, having described principles and objectives, it turns to implementation pathways (Part III). Thus, the means considered are guided by the objectives pursued.
1.3. Comparisons
Molecular nanotechnology resembles and overlaps with other fields, yet differs substantially. A discussion of resemblances can illustrate the applicability of existing knowledge and emerging developments; a discussion of differences can warn of false analogies and consequent misunderstandings.
Table 1.3 compares several existing production processes-conventional fabrication, microfabrication, solution-phase chemistry, and biochemistry-with molecular manufacturing. The following sections provide a more detailed comparison of molecular manufacturing and molecular nanotechnology with these other processes and their products. Appendix B focuses on areas of these fields having special relevance molecular manufacturing; the present section makes broad comparisons to the mainstream.
1.3.1. Conventional fabrication and mechanical engineering
a. Similarities: components, systems, controlled motion, manufacturing. Many mechanical engineering concepts apply directly to nanomechanical systems. As shown in Chapters 3 to 6, methods based on classical mechanics suffice for much of the required analysis. As shown in Chapters 9 to 11, beams, shafts, bearings, gears, motors, and the like can all be constructed on a nanometer scale to serve familiar mechanical functions. As a consequence, macromechanical
Table 1.3. Typical characteristics of conventional machining, micromachining, solution-phase chemistry, biochemistry, and molecular manufacturing.
| Characteristic | Conventional fabrication | Micro- fabrication | Solution chemistry | Bio- chemistry | Molecular manufacturing |
|---|---|---|---|---|---|
| Molecular precision? | no | no | yes | yes | yes |
| Positional control? | yes | yes | no | partial | yes |
| Typical feature scale | |||||
| Typical product scale | |||||
| Typical defect rate | |||||
| Typical cycle times | |||||
| Products described by | materials | materials | atoms | monomer | atoms |
Product scale, defect rates, and cycle times vary widely from process to process within most of these families; feature scale varies widely within the first two.
b The defect rate listed for biochemistry corresponds to high-reliability DNA replication processes that include kinetic proofreading (Watson et al., 1987); most biochemical defect rates are higher. All rates are on a per-component basis.
engineering and nanomechanical engineering share many design issues and analytical techniques.
Both molecular and conventional manufacturing systems use machines to perform planned patterns of motion: they shape, move, and join components to build complex three-dimensional structures. Systems of both kinds can manufacture machines, including machines used for manufacturing (Chapter 14).
b. Differences: scale, molecular phenomena. Despite these similarities, nanomechanical engineering forms a distinct field. The familiar model of objects as made of homogeneous materials, though still useful (Chapter 9), often must be replaced by models that treat objects as sets of bonded atoms (Chapters 2 and 3). Thermally excited vibrations are of major importance, and quantum effects are sometimes significant (Chapters 5, 6, and 7). Further, nanomachines suffer from molecular damage mechanisms (Chapter 6); molecular phenomena permit (and demand) novel bearings (Chapter 10); scaling laws favor electrostatic over electromagnetic motors (Chapters 2 and 11); and the basic operations of manufacturing on a molecular scale are chemical transformations (Chapters 8 and 13). These transformations typically move the system between discrete states, leading to structures that are either exactly right or clearly wrong; this resembles digital logic more closely than it does metalworking.
1.3.2. Microfabrication and microtechnology
a. Similarities: small scale, electronic quantum effects. Microtechnology has enabled the fabrication of micron-scale mechanical devices. These share basic scaling properties with nanomechanical devices-and so, for example, electrostatic motors are preferred over electromagnetic motors in both microand nanotechnology (Section 2.4.3). Further, quantum electronic devices of kinds now being explored with microfabrication technologies may become targets for molecular manufacturing.
b. Differences: fabrication technology, scale, molecular phenomena. Microfabrication relies on technologies (photolithographic pattern definition, etching, deposition, diffusion) essentially unrelated to those of molecular manufacturing. In a sense these two fields are moving in opposite directions: microfabrication attempts to make bulk-material structures smaller despite fabrication irregularities; molecular manufacturing will emerge from attempts to make molecular structures larger without losing the atomic precision characteristic of stereospecific chemical synthesis. Making structures consisting of a few dozen preciselyarranged atoms seems unachievable using microfabrication, but is routine in chemical synthesis. The gears, bearings, and motors described in Chapters 10 and 11 differ in volume from their closest microfabricated counterparts by a factor of , and rely on molecular structures and phenomena in their operation.*
1.3.3. Solution-phase chemistry
a. Similarities: molecular structure, processes, fabrication. Chemical principles describe the basic steps of molecular manufacturing, since each consists of a chemical transformation. Chemical knowledge can help in evaluating the stability of products, and chemical research has produced the most useful models of the mechanical behavior of molecular objects. Organic chemistry is particularly relevant owing to the superiority of carbon-based structures for most mechanical applications. Fundamental chemical concepts such as bonding, strain, reaction rates, transition states, orbital symmetry, and steric hindrance are all applicable; familiar chemical entities such as alkanes, alkenes, aromatic rings, radicals, and carbenes are all useful. Solution-based organic synthesis can make precisely structured molecular objects; it has even been used to make molecular gears (Mislow, 1989), although of a sort having no obvious utility for nanomechanical engineering.3b. Differences: machine-phase systems, mechanosynthesis. The chief differences between the present subject and conventional chemistry stem from the properties of machine-phase systems and of mechanosynthesis. These have been summarized in Section 1.1.2 and are discussed at length in Chapter 8.
1.3.4. Biochemistry and molecular biology
a. Similarities: molecular machines, molecular systems. Molecular biology, like molecular nanotechnology, embraces the study of molecular machines and molecular machine systems. Ribosomes-like mechanisms in flexible molecular manufacturing systems-can be viewed as numerically controlled machine tools following a series of instructions to produce a complex product. Molecular biology and biochemistry stimulated the train of thought that led to the concept of molecular manufacturing (Drexler, 1981), and their techniques offer paths to the development of molecular manufacturing systems (Section 15.2).
b. Differences: materials, machine phase, general mechanosynthesis. Biology is a product of evolution rather than design, and molecular biologists study systems that differ greatly from the eutactic systems described here. Unlike molecular manufacturing systems, the molecular machines found in cells can synthesize only relatively small molecules and a stereotyped set of polymers; they cannot synthesize a broad class of diamondoid structures. Larger biological structures typically acquire their shapes through the action of weak forces ('hydrogen bonds, salt bridges, van der Waals attraction, hydrophobic forces). As a consequence of stronger bonding, the strength and modulus of diamondoid components can exceed those of biological structures by orders of magnitude. The bearings, gears, motors, and computers discussed in Part II are accordingly quite different from the bacterial flagellar motor, the actin-myosin system, systems of neurons, and so forth. Biological and nanomechanical systems are organized in fundamentally different ways. For example, cells rely on diffusion in a liquid phase-although they contain molecular machines, they are not machine-phase systems.*
1.4. The approach in this volume
1.4.1. Disciplinary range, level, and presentation
As Section 1.2 indicates, the study of molecular nanotechnology spans multiple disciplines. This circumstance has hampered both evaluation of the existing concepts and research aimed at extending and superseding them. One purpose4of the present volume is to assemble a large portion of the necessary core knowledge in a form that requires no specialized knowledge of the component disciplines. An effort has been made (and a glossary provided) to make key chemical concepts accessible to nonchemists, solid-state physics concepts accessible to nonphysicists, and so forth, assuming only a basic background in both chemistry and physics (and a willingness to skip past the occasional obscure observation aimed at a reader in a different discipline). The intended contribution of this work consists not in extending the frontiers of existing fields, but in combining their basic results to lay the foundations of a new field.
To facilitate understanding, several mathematical results in Part I are derived from fundamental principles. Many of these results appear in existing textbooks; others (so far as is known) are novel, being motivated by new questions. The exposition of these mathematical models includes an unusually large number of graphs that illustrate equations in the text; these are provided to facilitate design, which is a synthetic as well as an analytic process. In the analysis of a given system, a calculation based on an equation with a single set of parameter values frequently suffices. In synthesis, however, the designer usually wishes to understand how system properties will vary as controllable parameters are changed; for this, a graph can be more useful than a bare equation.
Different fields have applied different energy units to molecular-scale phenomena, including the kilocalorie per gram-mole of items per single item) and the kilojoule per gram-mole of items per single item) of chemistry, and the electron-volt of physics. The standard SI unit of energy, of course, is the joule itself. To avoid allusions to hypothetical moles of identical systems or to electrons not involved in the problem, and (more important) to enable mechanical calculations involving force, work, kinetic energy, and so forth to proceed without frequent unit conversions, this volume adheres to the joule as the unit of energy. The attojoule and milli-attojoule ) are convenient fractional units.
1.4.2. Levels of abstraction and approximation
In an ideal world, chemists would be able to predict the behavior of molecules by applying quantum electrodynamics (QED) to suitably defined assemblages of nuclei and electrons, and engineers would be able to predict automotive performance in the same manner. In this regard, the world is far from ideal. Although no experiment has yet shown an imperfection in QED as a description of the properties of ordinary matter (setting nuclear and gravitational interactions aside), it is computationally intractable as a description even of small molecules. In the real world, chemists and engineers describe systems using a hierarchy of levels of abstraction and approximation (Table 1.4). It is worth surveying this hierarchy because it provides a framework for practical analysis.
As discussed in Chapter 3, the most rigorous models ordinarily used by chemists apply initio molecular orbital methods; these approximate the Schrödinger equation, which approximates the Dirac equation, which approximates QED, which approximates the unknown exact, universal laws of physics. In describing the mechanical properties of large molecular structures, however, chemists abandon molecular orbital methods in favor of molecular mechanics
Table 1.4. Levels of abstraction and approximation in molecular systems engineering.

Schrödinger equation
Quantum electrodynamics
methods of more limited applicability but lower computational cost; these too are discussed in Chapter 3.
At the upper levels of the hierarchy, engineers set objectives in terms of system behavior and use these objectives to determine requirements for subsystem behavior (this can proceed through several layers of subsystems). Systems are commonly analyzed in terms of subsystem capabilities, which are analyzed in terms of lumped-component models, which in turn are analyzed in terms of continuum models. For example, a modern computer is described by its subsystems-processor, memory, disk, bus, cooling, power supply, and so forth. A processor (give or take some intermediate levels) is described as an interconnected network of discrete transistors and other lumped components. Transistors are described by continuum models that consider gate geometries, dopant distributions, electron transport, and so forth. Individual atoms and electrons are neglected in describing transistors, and one never describes a computer by describing electron transport within transistors.
Nanomechanical systems are subject to a similar analysis, describing systemlevel objectives served by subsystem capabilities implemented using lumped components. Continuum models, however, become problematic on the nanometer scale. Chapter 9 develops bounded continuum models that take sufficient account of surface effects to enable the analysis of a broad range of nanomechanical designs in less than atomic detail. To design the smallest devices, however, detailed molecular mechanics models are necessary, and to provide a firstprinciples analysis of a mechanochemical process, there is no substitute for molecular orbital methods.
1.4.3. Scope and assumptions
The present volume adheres to design constraints that may not limit future engineering practice. Each constraint excludes possibilities that are presently difficult to analyze, but that may prove both feasible and desirable. The following

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Figure 1.4. Periodic table of the atoms, with cells shaded to indicate those of greatest importance to nanomechanical design: hydrogen , carbon , nitrogen , oxygen , fluorine , silicon , phosphorus , sulfur , and chlorine . Other elements have applications, but few of the structures discussed in the following chapters contain them. [H is more often placed above lithium ( than above , but hydrocarbons resemble the stable fluorocarbons far more than they do the reactive lithiocarbons. Like F, H is only one electron short of a closed-shell configuration.] b. No nanoelectronic devices. On a macroscale, mechanical systems are quite distinct from electronic systems: they involve the motion of materials, rather than of electrons and electromagnetic fields. On a nanoscale, mechanical motions are identified with the displacements of nuclei, but electronic activity can cause such motions. Nevertheless, many systems (e.g., the bearing in Figure 1.1) can be described by molecular mechanics models that take no explicit account of electronic degrees of freedom, instead subsuming them into a potential energy function (Chapter 3) defined in terms of the positions of nuclei. This volume focuses on mechanical systems of this sort.
Some systems are strongly electronic in character, relying on changes of electronic state to change other electronic states, with the associated nuclear motions being of small amplitude. Molecular nanotechnologies will surely include nanometer-scale electronic devices exploiting quantum phenomena to achieve (for example) switching and computation. Research relevant to this class of devices is already in progress.
Although nanoelectronic devices are likely to be important products of molecular manufacturing, they fall beyond the scope of the present work. There are several reasons for this. First, the analytical methods required for quantum electronics differ from those required for molecular machinery; including them would have made this volume larger and later. Second, these analytical methods require approximations that frequently render the results dubious, reducing their value as a premise for further reasoning (the existence in 1992 of at least three competing classes of theories to explain high-temperature superconductivity, observed and studied in cuprates since 1987, suggests the magnitude of the difficulties). Finally, while nanomechanical devices can build nanoelectronic devices, the latter cannot return the favor; thus, nanomechanical devices are in a technological sense more fundamental. Accordingly, the nanocomputers discussed in Chapter 12 are based on mechanical devices, although electronic devices will surely permit greater speed.
Chapter 11 briefly discusses nanoscale electromechanical systems. These use conductors, insulators, and tunneling junctions as components of motors, actuators, and switches. Quantum phenomena are important in nanoscale electromechanical systems, but the gross results (switching, interconversion of electrical and mechanical energy) do not rely on subtle quantum effects. Finally, despite their likely utility, machine-phase electrochemical processes are mentioned only in passing.
c. Machine-phase chemistry. Because it can guide reactions (and can avoid most competing reactions) by tightly constraining molecular motions, machinephase chemistry can be simpler, in some respects, than is solution-phase chemistry. Mechanosynthesis and other operations can be conducted by systems of molecular machines immersed in a solution environment, and there are sometimes advantages to doing so. These less controlled, more complex chemical systems fall beyond the scope of the present work, save for the discussion of implementation strategies in Part III.
d. Room temperature processes. Reduced temperatures decrease thermally excited displacements (Chapter 5), thermal damage rates (Chapter 6), and phonon-mediated drag (Chapter 7). The opportunities and problems presented by low-temperature systems are nonetheless not explored in the present work. Operation at high temperatures is desirable in some circumstances, and can facilitate both desired and undesired chemical reactions, but discussions of the associated technological opportunities and problems are likewise omitted.
e. No photochemistry. Photochemical damage mechanisms are discussed in Section 6.5. Design of molecular machines for photochemical damage resistance is an important challenge, but for simplicity the following will instead assume that devices operate in optically shielded environments. Deliberate use of photochemistry is mentioned only in passing.
f. The single-point failure assumption. The design of small components that can tolerate atomic-scale damage and defects is worthy of attention, but the following will instead assume that any such flaw causes component-level (and usually subsystem-level) failure, unless the components are relatively large. The use of redundant systems to achieve damage tolerance is proposed and analyzed only at higher levels of system organization. Diverse damage mechanisms are reviewed and modeled in Chapter 6.
1.4.4. Objectives and nonobjectives
Because this volume is a work of theoretical applied science (Appendix A), some objectives appropriate to pure and applied experimental studies, or to pure theoretical studies, are inappropriate here. The following paragraphs outline both familiar objectives that are neglected and less-familiar objectives that are pursued. Appendix A describes these issues in more depth.
a. Not describing new principles and natural phenomena. An enormous literature describes new natural phenomena, but this volume describes the implications of known phenomena for new technologies.
b. Seldom formulating exact physical models. In analyzing functional systems, estimates should be accurate or conservative, but need not be exact. The goal is to gain a quantitative understanding of the relationship between structure and function, not to formulate exact physical models for their own sakes.
c. Seldom describing immediate objectives. Most of the scientific literature discusses either past achievements or objectives achievable using existing techniques. Few publications examine objectives that require substantial preparatory development (space science and high-energy physics produce many of the exceptions). Constructing the physical systems discussed in Chapters 8-14 and 16 , however, is well beyond current capabilities. In this volume, only Chapter 15 describes objectives suited to the constraints of present laboratory technique. Our ability to model has outstripped our ability to make, and these studies exploit that fact to analyze ambitious objectives.
d. Not portraying specific future developments. This volume describes systems that can deliver performance orders of magnitude beyond that possible with current technology. Nonetheless, as research in this area expands, better designs will in most instances supersede these systems prior to their realization.
Accordingly, they should be regarded not as portrayals of specific future developments, but merely as examples of what can be done.
e. Seldom seeking an optimal design in the conventional sense. In mature fields of technology, competitive pressures encourage a search for designs that are nearly optimal. In the exploratory phase of design, however, the more modest goal of workability suffices. A design can depart from optimality either (1) by being overdesigned and inefficient, or (2) by being underdesigned and unreliable. Here, (1) is acceptable, (2) is not.
f. Seldom specifying complete detail in complex systems. Given an estab-
lished technology base, a designer can often describe a system at a high level of abstraction (Section 1.4.2) with confidence that compatible sets of components can be specified before placing the unit into production. This is easier if the product can be overdesigned and inefficient, because margins of safety in the initial design can then compensate for uncertainties in component performance. The approach pursued in this volume amounts to the analytical (as distinct from physical) development of a technology base with capabilities known within some error margins. The later chapters describe systems at various levels of abstraction, likewise using margins of safety to compensate for uncertainties in component performance.g. Favoring false-negative over false-positive errors in analysis. In modeling and analyzing proposed designs, one would ideally distinguish workable from unworkable designs with perfect accuracy. But since models are never exact and accurate, errors of some sort are inevitable. These errors can be of two kinds: False-positive evaluations wrongly accept an unworkable design; falsenegative evaluations wrongly reject a workable design. In exploring a new domain of technology, conclusions regarding the feasibility of systems rest on conclusions regarding the feasibility of subsystems, forming a hierarchical structure of analysis that can have several layers. A substantial rate of false-positive assessments at a subsystem level would make false-positive assessments at the system level quite likely: designs that rely on unworkable subsystems will not work, and may be beyond repair. False-negative assessments, in contrast, are relatively benign. Correcting a false-negative assessment of a rejected subsystem cannot invalidate the analysis of a system whose design omits that subsystem; indeed, correcting this error merely expands the list of workable designs. It is desirable to minimize errors of both kinds, but where uncertainties remain, it is better to bias analytical models and criteria toward safe, false-negative conclusions. This strategy guides the following chapters.
h. Describing technological systems of novel kinds and capabilities. The objective of this volume, then, is to provide an analytical framework adequate for designing nanomechanical systems (Chapters 3 to 10), and to begin to exploit that framework by designing systems capable of processing both information (mechanical nanocomputers) and matter (molecular manufacturing systems). A further objective is to show how these systems can be implemented, starting from our present technology base (Chapters 15 and 16). Achieving these objectives can define fruitful goals for experimentation, simulation, and software development, and can provide a better basis for understanding human capabilities and choices in the coming years.
1.5. Overview of following chapters
In a work of this length in a new field, the relationship among topics can easily become lost in a mass of detail. This section provides a chapter-by-chapter overview, describing relationships and indicating areas where a detailed analysis merely shows that a simpler analysis is sufficient.
1.5.1. Overview of Part I
Chapter 2 summarizes classical scaling laws for mechanical, electrical, and thermal systems, describing the magnitudes they imply for various physical parameters, and describing where and how these laws break down (requiring the substitution of molecular and quantum mechanical models). These relationships are provided for perspective and as an aid in making preliminary estimates of physical quantities. They are seldom used in later chapters, where most calculations proceed directly from physical principles, rather than from a scaling analysis.
Chapter 3 provides an overview of molecular and intersurface potential energy functions, describing in some detail the molecular mechanics models that later chapters apply to the description of molecular machines. The concepts developed in this chapter are fundamental to much of what follows, since the potential energy function of a molecular system provides the basis for an essentially complete description of its mechanical properties. Its chief conclusion is that the limitations and inaccuracies of molecular mechanics models, although a serious handicap in solution-phase chemistry, are compatible with the use of these models in designing certain classes of molecular machinery.
Chapter 4 provides an overview of various models of molecular dynamics and describes the basis for the choice of models made in later chapters. Its chief conclusion is that classical mechanics and classical statistical mechanics, with occasional quantum mechanical corrections, provide an adequate basis for analyzing most molecular mechanical systems operating at ordinary temperatures.
Chapter 5 examines several classes of mechanical structures and derives relationships that describe the positional uncertainties resulting from the combined effects of quantum mechanics and thermal excitation; later chapters use these results to calculate error rates. This is the most heavily mathematical chapter in the book, but it chiefly concludes that classical statistical mechanics adequately describes the positional uncertainties of nanometer-scale mechanical systems at ordinary temperatures. Only Eq. (5.4) is directly applied in later chapters.
Chapter 6 examines various processes that cause transitions among potential wells in a nanomechanical system, including transitions that cause errors and structural damage. It describes standard theoretical models used to calculate chemical reaction rates based on potential energy functions; these are applied later in analyzing molecular manufacturing processes. Regarding damaging transitions, the chapter concludes that systems at room temperature can be designed to limit rates of damage caused by thermal, mechanical, and photochemical effects to low enough levels that the overall rate of damage is chiefly determined by the background level of ionizing radiation. Damage caused by radiation imposes major constraints on the design of large systems.
Chapter 7 examines various processes that degrade mechanical energy into heat, causing frictional losses; among these are acoustic radiation, phonon scattering, shear-reflection drag, phonon viscosity, thermoelastic damping, nonisothermal compression of mobile components, and transitions among potential wells that vary with time. It develops a set of analytical models applied in later chapters to estimate the power dissipated by various nanomechanical devices.
Chapter 8 compares and contrasts the established capabilities of solution chemistry to those expected from mechanochemical systems, considering speed, efficiency, versatility, and reliability; it also examines an illustrative set of mechanochemical processes in detail. It concludes that a large set of mechanochemical reactions can be made extremely reliable, with error rates . (Although not strictly necessary, this degree of reliability substantially simplifies molecular manufacturing.) It also concludes that mechanochemical processes can be used to construct a wide range of diamondoid structures; this motivates the consideration of diamondoid components in Part II.
- Overall, Part I expends substantial effort in describing physical effects that are of negligible importance in nanomechanical design. In established fields of engineering, training and experience focus attention on the important physical effects; most trivial effects are automatically ignored. In surveying a new field, however, insignificant effects must often be examined before they can be recognized as insignificant.
1.5.2. Overview of Part II
Chapter 9 discusses nanoscale structural components and relates molecular mechanics models to descriptions based on bulk material properties. It develops the concept of a bounded continuum model that omits atomic detail while treating surfaces in a way that takes account of interatomic potentials. It concludes that diamondoid structures are desirable nanoscale components, that bounded continuum models can provide useful preliminary descriptions of component properties, and that a wide range of shapes can be constructed on a nanometer scale, despite the discrete nature of atoms and bonds.
Chapter 10 uses molecular mechanics models and analytical models developed in Part I to describe the mechanical properties and performance characteristics of active devices such as gears, bearings, and drive mechanisms. It concludes that structures on a several-nanometer scale can serve as mechanical devices of most classes familiar on the macroscopic scale, and that these nanomechanical devices can in many instances move with negligible static friction. Models from Chapter 7 are applied to describe energy dissipation from dynamic friction. The conditions for low-friction motion derived in this chapter are assumed as background in Chapters 11 to 14.
Chapter 11 describes various subsystems of intermediate size and complexity. These include devices capable of measuring and distinguishing molecular features, stiff drive mechanisms, fluid handling systems (including walls, seals, and vacuum pumps), cooling systems, and electromechanical devices such as motors and actuators. These subsystems and their capabilities are applied or assumed as background in Chapters 12 to 14.
Chapter 12 describes nanomechanical computer systems. It starts with mechanical digital logic systems, including logic gates, signal transmission channels, registers, and their integration into finite-state machines (with an analysis of thermally induced error rates). It then discusses carry chains, random access memory, tape-based storage, power supplies, clock distribution, information input and output to macroscopic systems, and overall system performance (volume, speed, and power dissipation). Its chief conclusion is that a 1000 MIPS computer can occupy less than one cubic micron and consume less than 0.1 microwatt of power.
Chapter 13 describes systems for converting impure feedstock solutions into diamondoid molecular objects, using molecular concentration and purification systems, followed by special-purpose mechanochemical systems (molecular mills) capable of performing repetitive operations efficiently, and by generalpurpose mechanochemical systems (i.e., molecular manipulators) capable of performing a complex series of operations under programmable control. A key conclusion is that, after an initial purification and ordering process, molecular assembly can be sufficiently reliable that cycles of inspection and rework to correct failures can be avoided. The conclusions of this chapter are directly exploited in the next.
Chapter 14 describes molecular manufacturing systems that use purification systems, mills, and manipulators to transform an impure feedstock solution into any one of a large set of macroscopic (kilogram-scale) products within a few thousand seconds. It focuses on systems integration and overall performance, and discusses a range of issues including design complexity and product cost.
- Part II describes diamondoid nanomechanical systems of increasing complexity, ending with a description of molecular manufacturing systems that can build complex diamondoid nanomechanical systems. This supports the nontrivial proposition that molecular manufacturing systems are feasible, given molecular manufacturing systems with which to build them. Part III discusses the implementation of molecular manufacturing systems in the absence of preexisting systems of the same kind. It begins from our present technology base.
1.5.3. Overview of Part III
Chapter 15 describes current capabilities for the design and fabrication of multinanometer scale molecular objects (by chemical and biochemical means) and discusses approaches for combining and extending these capabilities to enable the construction of larger and more complex systems. The major approaches considered are (1) protein engineering, (2) the engineering of proteinlike molecules designed to fold more predictably and stably, and (3) the development of mechanosynthetic systems that make use of atomic force microscope (AFM) technologies. The first two approaches would construct large molecular systems by Brownian assembly (self assembly) of smaller units; the AFM approach would construct them directly from small monomers. This chapter concludes that means can be developed for constructing molecular mechanical systems containing monomers, operating in a solution environment.
Chapter 16 describes a development pathway leading from small molecular mechanical systems operating in solution, through larger and better-isolated systems, to scale mechanisms able make complex diamondoid structures by means of mechanosynthesis. It proposes the use of externally generated pressure fluctuations to provide power and control to these intermediatetechnology devices, and the use of optically probed, environmentally modulated fluorescent molecules to enable prompt sensing of the results of attempted mechanosynthetic operations. This chapter concludes that accessible (though not necessarily easy) development paths lead from present capabilities to advanced molecular manufacturing.
- Part II begins and ends with long-term technologies, but Part III begins with current technologies. Readers more interested in short-term developments and practical foundations may wish to start with Part III. It is relatively independent of the rest, although it occasionally cites results from previous chapters.
1.5.4. Overview of Appendices
Appendix A discusses methodologies appropriate to the study of technological possibilities, comparing and contrasting them to the methodologies appropriate to the study of natural phenomena (i.e., standard scientific problems) and to the design of products for immediate implementation (i.e., standard engineering problems).
Appendix B discusses related work by other researchers. It returns to the fields of mechanical engineering, microtechnology, chemistry, and molecular biology (adding protein engineering and proximal probe technology), focusing on the work in these fields that has advanced furthest toward engineering complex molecular systems.
1.5.5. Open problems
Many chapters end with a discussion of open problems. These range from developing specific examples and analyses to developing major software systems and laboratory research programs. The discussions are not exhaustive, but are intended to highlight useful directions for research.
Part
Physical Principles
Classical Magnitudes and Scaling Laws
2.1. Overview
Most physical magnitudes characterizing nanoscale systems differ enormously from those familiar in macroscale systems. Some of these magnitudes can, however, be estimated by applying scaling laws to the values for macroscale systems. Although later chapters seldom use this approach, it can provide orientation, preliminary estimates, and a means for testing whether answers derived by more sophisticated methods are in fact reasonable.
The first of the following sections considers the role of engineering approximations in more detail (Section 2.2); the rest present scaling relationships based on classical continuum models and discuss how those relationships break down as a consequence of atomic-scale structure, mean-free-path effects, and quantum mechanical effects. Section 2.3 discusses mechanical systems, where many scaling laws are quite accurate on the nanoscale. Section 2.4 discusses electromagnetic systems, where many scaling laws fail dramatically on the nanoscale. Section 2.5 discusses thermal systems, where scaling laws have variable accuracy. Finally, Section 2.6 briefly describes how later chapters go beyond these simple models.
2.2. Approximation and classical continuum models
When used with caution, classical continuum models of nanoscale systems can be of substantial value in design and analysis. They represent the simplest level in a hierarchy of approximations of increasing accuracy, complexity, and difficulty.
Experience teaches the value of approximation in design. A typical design process starts with the generation and preliminary evaluation of many options, then selects a few options for further elaboration and evaluation, and finally settles on a detailed specification and analysis of a single preferred design. The first steps entail little commitment to a particular approach. The ease of exploring and comparing many qualitatively distinct approaches is at a premium, and drastic approximations often suffice to screen out the worst options. Even the final design and analysis does not require an exact calculation of physical behavior: approximations and compensating safety margins suffice. Accordingly, a design process can use different approximations at different stages, moving toward greater analytical accuracy and cost.
Approximation is inescapable because the most accurate physical models are computationally intractable. In macromechanical design, engineers employ approximations based on classical mechanics, neglecting quantum mechanics, the thermal excitation of mechanical motions, and the molecular structure of matter. Since macromechanical engineering blends into nanomechanical engineering with no clear line of demarcation, the approximations of macromechanical engineering offer a point of departure for exploring the nanomechanical realm. In some circumstances, these approximations (with a few adaptations) provide an adequate basis for the design and analysis of nanoscale systems. In a broader range of circumstances, they provide an adequate basis for exploring design options and for conducting a preliminary analysis. In a yet broader range of circumstances, they provide a crude description to which one can compare more sophisticated approximations.
2.3. Scaling of classical mechanical systems
Nanomechanical systems are fundamental to molecular manufacturing and are useful in many of its products and processes. The widespread use in chemistry of molecular mechanics approximations together with the classical equations of motion (Sections 3.3, 4.2.3a) indicates the utility of describing nanoscale mechanical systems in terms of classical mechanics. This section describes scaling laws and magnitudes with the added approximation of continuous media.
2.3.1. Basic assumptions
The following discussion considers mechanical systems, neglecting fields and currents. Like later sections, it examines how different physical magnitudes depend on the size of a system (defined by a length parameter ) if all shape parameters and material properties (e.g., strengths, moduli, densities, coefficients of friction) are held constant.
A description of scaling laws must begin with choices that determine the scaling of dynamical variables. A natural choice is that of constant stress. This implies scale-independent elastic deformation, and hence scale-independent shape; since it results in scale-independent speeds, it also implies constancy of the space-time shapes describing the trajectories of moving parts. Some exemplar calculations are provided, based on material properties like those of diamond (Table 9.1): density ; Young's modulus ; and a low working stress ( times tensile strength) . This choice of materials often yields large parameter values (for speeds, accelerations, etc.) relative to those characteristic of more familiar engineering materials.
2.3.2. Magnitudes and scaling
Given constancy of stress and material strength, both the strength of a structure and the force it exerts scale with its cross-sectional area
Nanoscale devices accordingly exert only small forces: a stress of equals , or . Stiffness in shear, like stretching stiffness, depends on both area and length
and varies less rapidly with scale; a cubic nanometer block of has a stretching stiffness of . The bending stiffness of a rod scales in the same way
Given the above scaling relationships, the magnitude of the deformation under load
is proportional to scale, and hence the shape of deformed structures is scale invariant.
The assumption of constant density makes mass scale with volume,
and the mass of a cubic nanometer block of density equals .
The above expressions yield the scaling relationship
A cubic-nanometer mass subject to a net force equaling the above working stress applied to a square nanometer experiences an acceleration of . Accelerations in nanomechanisms commonly are large by macroscopic standards, but aside from special cases (such as transient acceleration during impact and steady acceleration in a small flywheel) they rarely approach the value just calculated. (Terrestrial gravitational accelerations and stresses usually have negligible effects on nanomechanisms.)
Modulus and density determine the acoustic speed, a scale-independent parameter [along a slim rod, the speed is ; in bulk material, somewhat higher]. The vibrational frequencies of a mechanical system are proportional to the acoustic transit time
The acoustic speed in diamond is . Some vibrational modes are more conveniently described in terms of lumped parameters of stiffness and mass,
but the scaling relationship is the same. The stiffness and mass associated with a cubic nanometer block yield a vibrational frequency characteristic of a stiff, nanometer-scale object: .
Characteristic times are inversely proportional to characteristic frequencies
The speed of mechanical motions is constrained by strength and density. Its scaling can be derived from the above expressions
A characteristic speed (only seldom exceeded in practical mechanisms) is that at which a flywheel in the form of a slim hoop is subject to the chosen working stress as a result of its mass and centripetal acceleration. This occurs when (with the assumed and ). Most mechanical motions considered in this volume, however, have speeds between 0.001 and .
The frequencies characteristic of mechanical motions scale with transit times
These frequencies scale in the same manner as vibrational frequencies, hence the assumption of constant stress leaves frequency ratios as scale invariants. At the above characteristic speed, crossing a distance takes ; the large speed makes this shorter than the motion times anticipated in typical nanomechanisms. A modest speed, however, still yields a transit time of only , indicating that nanomechanisms can operate at frequencies typical of modern micron-scale electronic devices.
The above expressions yield relationships for the scaling of mechanical power
and mechanical power density
A force and a volume yield a power of and a power density of (at a speed of ) or and (at a speed of ). The combination of strong materials and small devices promises mechanical systems of extraordinarily high power density, even at low speeds (an example of a mechanical power density is the power transmitted by a gear divided by its volume).
Most mechanical systems use bearings to support moving parts. Macromechanical systems frequently use liquid lubricants, but (as noted by Feynman, 1961), this poses problems on a small scale. The above scaling law ordinarily holds speeds and stresses constant, but reducing the thickness of the lubricant layer increases shear rates and hence viscous shear stresses:
In Newtonian fluids, shear stress is proportional to shear rate. Molecular simulations indicate that liquids can remain nearly Newtonian at shear rates in excess of across a layer (e.g., in the calculations of Ashurst and Hoover, 1975), but they depart from bulk viscosity (or even from liquid behavior) when film thicknesses are less than 10 molecular diameters (Israelachvili, 1992; Schoen et al., 1989), owing to interface-induced alterations in liquid structure. Feynman suggested the use of low-viscosity lubricants (such as kerosene) for micromechanisms (Feynman, 1961); from the perspective of a typical nanomechanism, however, kerosene is better regarded as a collection of bulky molecular objects than as a liquid. If one nonetheless applies the classical approximation to a film of low-viscosity fluid , the viscous shear stress at a speed of is ; the shear stress at a speed of , is still large, dissipating energy at a rate of .
The problems of liquid lubrication motivate consideration of dry bearings (as suggested by Feynman, 1961). Assuming a constant coefficient of friction,
and both stresses and speeds are once again scale-independent. The frictional power,
is proportional to the total power, implying scale-independent mechanical efficiencies. In light of engineering experience, however, the use of dry bearings would seem to present problems (as it has in silicon micromachine research). Without lubrication, efficiencies may be low, and static friction often causes jamming and vibration.
A yet more serious problem for unlubricated systems would seem to be wear. Assuming constant interfacial stresses and speeds (as implied by the above scaling relationships), the anticipated surface erosion rate is independent of scale. Assuming that wear life is determined by the time required to produce a certain fractional change in shape,
and a centimeter-scale part having a ten-year lifetime would be expected to have a lifetime if scaled to nanometer dimensions.
Design and analysis have shown, however, that dry bearings with atomically precise surfaces need not suffer these problems. As shown in Chapters 6, 7, and 10 , dynamic friction can be low, and both static friction and wear can be made negligible. The scaling laws applicable to such bearings are compatible with the constant-stress, constant-speed expressions derived previously.
2.3.3. Major corrections
The above scaling relationships treat matter as a continuum with bulk values of strength, modulus, and so forth. They readily yield results for the behavior of iron bars scaled to a length of , although such results are meaningless because a single atom of iron is over in diameter. They also neglect the influence of surfaces on mechanical properties (Section 9.4), and give (at best) crude estimates regarding small components, in which some dimensions may be only one or a few atomic diameters.
Aside from the molecular structure of matter, major corrections to the results suggested by these scaling laws include uncertainties in position and velocity resulting from statistical and quantum mechanics (examined in detail in Chapter 5). Thermal excitation superimposes random velocities on those intended by the designer. These random velocities depend on scale, such that
where the thermal energy measures the characteristic energy in a single degree of freedom, not in the object as a whole. For , the mean thermal speed of a cubic nanometer object at is . Random thermal velocities (commonly occurring in vibrational modes) often exceed the velocities imposed by planned operations, and cannot be ignored in analyzing nanomechanical systems.
Quantum mechanical uncertainties in position and momentum are parallel to statistical mechanical uncertainties in their effects on nanomechanical systems. The importance of quantum mechanical effects in vibrating systems depends on the ratio of the characteristic quantum energy ( , the quantum of vibrational energy in a harmonic oscillator of angular frequency ) and the characteristic thermal energy ( , the mean energy of a thermally excited harmonic oscillator at a temperature , if ). The ratio varies directly with the frequency of vibration, that is, as . An object of cubic nanometer size with has . The associated quantum mechanical effects (e.g., on positional uncertainty) are smaller than the classical thermal effects, but still significant (see Figure 5.2).
2.4. Scaling of classical electromagnetic systems
2.4.1. Basic assumptions
In considering the scaling of electromagnetic systems, it is convenient to assume that electrostatic field strengths (and hence electrostatic stresses) are independent of scale. With this assumption, the above constant-stress, constant-speed scaling laws for mechanical systems continue to hold for electromechanical systems, so long as magnetic forces are neglected. The onset of strong field-emission currents from conductors limits the electrostatic field strength permissible at the negative electrode of a nanoscale system; values of can readily be tolerated (Section 11.6.2).
2.4.2. Major corrections
Chapter 11 describes several nanometer scale electromechanical systems, requiring consideration of the electrical conductivity of fine wires and of insulating layers thin enough to make tunneling a significant mechanism of electron transport. These phenomena are sometimes (within an expanding range of conditions) understood well enough to permit design calculations.
Corrections to classical continuum models are more important in electromagnetic systems than in mechanical systems: quantum effects, for example, become dominant and at small scales can render classical continuum models useless even as crude approximations. Electromagnetic systems on a nanometer scale commonly have extremely high frequencies, yielding large values of . Molecules undergoing electronic transitions typically absorb and emit light in the visible to ultraviolet range, rather than the infrared range characteristic of thermal excitation at room temperature. The mass of an electron is less than that of the lightest atom, hence for comparable confining energy barriers, electron wave functions are more diffuse and permit longer-range tunneling. At high frequencies, the inertial effects of electron mass become significant, but these are neglected in the usual macroscopic expressions for electrical circuits. Accordingly, many of the following classical continuum scaling relationships fail in nanoscale systems. The assumption of scale-independent electrostatic field strengths itself fails in the opposite direction, when scaling up from the nanoscale to the macroscale: the resulting large voltages introduce additional modes of electrical breakdown. In small structures, the discrete size of the electronic charge unit, , disrupts the smooth scaling of classical electrostatic relationships (Section 11.7.2c).
2.4.3. Magnitudes and scaling: steady-state systems
Given a scale-invariant electrostatic field strength,
At a field strength of , a one nanometer distance yields a potential difference. A scale-invariant field strength implies a force proportional to area,
and a field between two charged surfaces yields an electrostatic force of .
Assuming a constant resistivity,
and a cubic nanometer block with the resistivity of copper would have a resistance of . This yields an expression for the scaling of currents,
which leaves current density constant. In present microelectronics work, current densities in aluminum interconnections are limited to or less by electromigration, which redistributes metal atoms and eventually interrupts circuit continuity (Mead and Conway, 1980). This current density equals (as discussed in Section 11.6.1b, however, present electromigration limits are unlikely to apply to well-designed eutactic conductors).
For field emission into free space, current density depends on surface properties and the electrostatic field intensity, hence
and field emission currents scale with ohmic currents. Where surfaces are close enough together for tunneling to occur from conductor to conductor, rather than from conductor to free space, this scaling relationship breaks down.
With constant field strength, electrostatic energy scales with volume:
A field with a strength of has an energy density of per cubic nanometer .
Scaling of capacitance follows from the above,
and is independent of assumptions regarding field strength. The calculated capacitance per square nanometer of a vacuum capacitor with parallel plates separated by is ; note, however, that electron tunneling causes substantial conduction through an insulating layer this thin.
In electromechanical systems dominated by electrostatic forces,
and
These scaling laws are identical to those for mechanical power and power density. Like them, they suggest high power densities for small devices (see Section 11.7).
The power density caused by resistive losses scales differently, given the above current density:
The current density needed to power an electrostatic motor, however, scales differently from that derived from a constant-field scaling analysis. In an electrostatic motor, surfaces are charged and discharged with a certain frequency, hence
and the resistive power losses climb sharply with decreasing scale:
Accordingly, the efficiency of electrostatic motors decreases with decreasing scale. The absence of long conducting paths (like those in electromagnets) makes resistive losses smaller to begin with, however, and a detailed examination (Section 11.7) shows that efficiencies remain high in absolute terms for motors of submicron scale. The above relationships show that electromechanical systems cannot be scaled in the simple manner suggested for purely mechanical systems, even in the classical continuum approximation.
Electromagnets are far less attractive for nanoscale systems, since
At a distance of from a conductor carrying a current of , the field strength is . The corresponding forces,
are minute in nanoscale systems: two parallel, long segments of conductor, separated by and carrying , interact with a force of . This is 14 orders of magnitude smaller than the strength of a typical covalent bond and 11 orders of magnitude smaller than the characteristic electrostatic force just calculated. Magnetic forces between nanoscale current elements are usually negligible. Magnetic fields generated by magnetic materials, in contrast, are independent of scale: forces, energies, and so forth follow the scaling laws described for constant-field electrostatic systems. Nanoscale current elements interacting with fixed magnetic fields can produce more significant (though still small) forces: a long segment of conductor carrying a current experiences a force of up to when immersed in a field.
The magnetic field energy of a nanoscale current element is small:
The scaling of inductance can be derived from the above, but is independent of assumptions regarding the scaling of currents and magnetic field strengths:
The inductance per nanometer of length for a fictitious solenoid with a cross sectional area and one turn per nanometer of length would be .
2.4.4. Magnitudes and scaling: time-varying systems
In systems with time-varying currents and fields, skin-depth effects increase resistance at high frequencies; these effects complicate scaling relationships and are ignored here. The following simplified relationships are included chiefly to illustrate trends and magnitudes that preclude the scaling of classical AC circuits into the nanometer size regime.
For circuits,
Combining the characteristic resistance and inductance calculated above yields an time constant of . This time constant is nonphysical: it is, for example, short compared to the electron relaxation time in a typical metal at room temperature ( . In reality, current decays more slowly because of electron inertia (which has effects broadly similar to those of inductance) and because of the related effect of finite electron relaxation time.
With the approximation of scale-independent resistivity,
This implies that the time required for a capacitor to discharge through a resistor in a pure circuit is independent of scale; with the scale dependence of the time constant, however, a structure with fixed proportions can change from a nearly pure circuit (if built on a small scale) to a nearly pure circuit (if built on a large scale). The nanometer-scale time constant indicated by this expression is , but this result is nonphysical because it neglects the effects of electron inertia and relaxation time.
The product defines an oscillation frequency
The characteristic inductance and capacitance calculated above would yield an circuit with an angular frequency of . Alternatively, in structures such as waveguides,
To propagate in a hypothetical waveguide in diameter, an electromagnetic wave would require a frequency of or more. Even the lower of the two frequencies just mentioned corresponds to quanta with an energy of , that is, to photons in the -ray range with energies of . These frequencies and energies are inconsistent with physical circuits and waveguides (metals are transparent to -rays, electrons are stripped from molecules at energies well below , etc.). Quantum effects and electron inertia make Eq. (2.38) inapplicable in the nanometer range.
Scale also affects the quality of an oscillator:
Since is a measure of the damping time relative to the oscillation time, small circuits will be heavily damped unless nonclassical effects intervene.
Where the following chapters consider electromagnetic systems at all, they describe systems with currents and fields that are slowly varying by the relevant standards. High-frequency quantum electronic devices, though undoubtedly of great importance, are not discussed here.
2.5. Scaling of classical thermal systems
2.5.1. Basic assumptions
The classical continuum model assumes that volumetric heat capacities and thermal conductivities are independent of scale. Since heat flows in nanomechanical systems are typically a side effect of other physical processes, no independent assumptions are made regarding their scaling laws.
2.5.2. Major corrections
Classical, diffusive models for heat flow in solids can break down in several ways. On sufficiently small scales (which can be macroscopic for crystals at low temperatures) thermal energy is transferred ballistically by phonons for which the mean free path would, in the absence of bounding surfaces, exceed the dimensions of the structure in question. In the ballistic transport regime, interfacial properties analogous to optical reflectivity and emissivity become significant. Radiative heat flow is altered when the separation of surfaces becomes small compared to the characteristic wavelength of blackbody radiation, owing to coupling of nonradiative electromagnetic modes in the surfaces. In gases, separation of surfaces by less than a mean free path again modifies conductivity. The following assumes classical thermal diffusion, which can be a good approximation for liquids and for solids of low thermal conductivity, even on scales approaching the nanometer range.
2.5.3. Magnitudes and scaling
With a scale-independent volumetric heat capacity,
A cubic nanometer volume of a material with a (typical) volumetric heat capacity of has a heat capacity of .
Thermal conductance scales like electrical conductance, with
and a cubic nanometer of material with a (fairly typical) thermal conductivity of has a thermal conductance of .
Characteristic times for thermal equilibration follow from these relationships, yielding
For a cubic nanometer block separated from a heat sink by a thermal path with a conductance of , the calculated thermal time constant is , which is comparable to the acoustic transit time. (In an insulator, a calculated thermal time constant approaching the acoustic transit time signals a breakdown of the diffusive model for transport of thermal energy and the need for a model accounting for ballistic transport; in the fully ballistic regime, time constants scale in proportion to , and thermal energy moves at the speed of sound.)
The scaling relationship for frictional power dissipation, Eq. (2.16), implies a scaling law for the temperature elevation of a device in thermal contact with its environment,
This indicates that nanomechanical systems are more nearly isothermal than analogous systems of macroscopic size.
Table 2.1. Summary of classical continuum scaling laws.
| Physical quantity | Scaling exponent | Typical magnitude | Scaling accuracy |
|---|---|---|---|
| Area | 2 | Definitional | |
| Force, strength | 2 | Good | |
| Stiffness | 1 | Good | |
| Deformation | 1 | Good | |
| Mass | 3 | Good | |
| Acceleration | -1 | Good | |
| Vibrational frequency | -1 | Good | |
| Stress-limited speed | 0 | Good | |
| Motion time | -1 | Good | |
| Power | 2 | Good | |
| Power density | -1 | Good | |
| Viscous stress | -1 | Moderate to poor | |
| Frictional force | 2 | Moderate to inapplicable | |
| Wear life | 1 | Moderate to inapplicable | |
| Thermal speed | Good | ||
| Voltage | 1 | Good at small scales | |
| Electrostatic force | 2 | Good at small scales | |
| Resistance | -1 | Moderate to poor | |
| Current | 2 | Moderate to poor | |
| Electrostatic energy | 3 | Good at small scales | |
| Capacitance | 1 | Good | |
| Magnetic field | 1 | Good | |
| Magnetic force | 4 | Good | |
| Inductance | 1 | Good | |
| Inductive time constant | 2 | Bad | |
| Capacitive time constant | 0 | - | Moderate to poor |
| Elect. oscill. frequency | -1 | Bad | |
| Oscillator Q | 1 | - | Moderate to poor |
| Heat capacity | 3 | Good | |
| Thermal conductance | 1 | Good to poor | |
| Thermal time constant | 2 | Good to poor |
Values included only to illustrate the failure of the scaling law.
b Values omitted; realistic geometries would require several arbitrary parameters.
2.6. Beyond classical continuum models
This chapter has described the scaling laws implied by classical continuum models for mechanical, electromagnetic, and thermal systems, together with the magnitudes they suggest for the physical parameters of nanometer scale systems. It has also considered limits to the validity of these models, imposed by statistical mechanics, quantum mechanics, the molecular structure of matter, and so forth. Different classical models fail at different length scales, with the most dramatic failures appearing in AC electrical circuits.
The following chapters go beyond classical continuum models. Chapters 3 and 4 examine models of molecular structure, dynamics, and statistical mechanics from a nanomechanical systems perspective. Chapters 5 and 6 examine the combined effects of quantum and statistical mechanics on nanomechanical systems, first analyzing positional uncertainty in systems subject to a restoring force, and then analyzing the rates of transitions, errors, and damage in systems that can settle in alternative states. Chapter 7 examines mechanisms for energy dissipation. These chapters provide a foundation for analyzing specific nanomechanical systems. Later chapters examine not only nanomechanical systems, but a few specific electrical and fluid systems; where analysis of the latter must go beyond classical continuum approximations, the needed principles are discussed in context.
2.7. Conclusions
The accuracy of classical continuum models and scaling laws to nanoscale systems depends on the physical phenomena considered. It is low for electromagnetic systems with small calculated time constants, reasonably good for thermal systems and slowly varying electromagnetic systems, and often excellent for purely mechanical systems, provided that the component dimensions substantially exceed atomic dimensions. Scaling principles indicate that mechanical components can operate at high frequencies, accelerations, and power densities. The adverse scaling of wear lifetimes suggests that bearings are a special concern. Later chapters support these expectations regarding frequency, acceleration, and power density; Chapter 10 describes suitable bearings. Table 2.1 summarizes many of the relationships discussed in this chapter.
Potential Energy Surfaces
3.1. Overview
The concept of a molecular potential energy surface (PES) is fundamental to practical models of molecular structure and dynamics. The PES describes the potential energy in terms of the molecular geometry, which in turn is defined by the positions of the atomic nuclei. In the classical approximation, molecular motions are determined by forces (and hence accelerations) corresponding to gradients of the PES, and equilibrium molecular structures correspond to minima of the PES. The term potential energy surface stems from a visualization in which a potential energy function in dimensions (that is, in configuration space) is described as a surface in dimensions, with energy corresponding to height. (When , the visualization is necessarily somewhat nonvisual.)
The significant features of a PES are its potential wells and the passes (termed ) between them. A point representing the state of a stiff, stable structure resides in a well with steep walls (i.e., with a large second derivative of the energy, the basis of stiffness, for all displacements) and no low, accessible cols leading to alternative wells (i.e., no low-energy routes to other structures, the basis of stability). A point representing the initial state of a chemically reactive structure, in contrast, resides in a well linked to another well by a col of accessible height. A point representing a mobile nanomechanical component commonly moves in a well with a long, level floor. Molecular systems in which all transitions occur between distinct potential wells resemble transistor systems in which all transitions occur between distinct logic states: in both instances, the application of correct design principles can yield reliable behavior even though other systems subject to the same physical principles behave erratically.
Physicists and chemists describe molecular behavior using a hierarchy of approximations of varying accuracy; the PES concept itself is one such approximation. The following sections move from extremely accurate but impractical theories to less accurate but more useful approximations. Section 3.2 discusses both exact quantum mechanical theories and chemically useful approximations to them. Section 3.3 discusses molecular mechanics methods at some length: for nanomechanical engineering, approaches like those described in Sections 3.3 to 3.5 often provide the most useful approximations to the PES, from the standpoint both of accuracy and of computational feasibility. (A discussion of the accuracy of these methods relative to different requirements appears in Section 4.4.) Section 3.4 discusses specialized potential energy functions describing chemical reactions, and Section 3.5 discusses models of the interaction energy of large objects, neglecting atomic detail. Most of the topics introduced in this chapter are discussed at length in the literature; suggestions for further reading are appended.
3.2. Quantum theory and approximations
The analysis of nanomechanical systems requires models describing the behavior of molecular scale systems. The appropriate models more closely resemble those used in chemistry and materials science than (for example) those used in particle physics. For perspective, however, it may be useful to view the hierarchy of approximations from the heights of modern physical theory before moving into the domain of practical calculation. Note that the following overview of quantum mechanics is offered chiefly for perspective; although some analytical models in this volume describe quantum mechanical effects, none makes direct use of wave equations or of the mathematical apparatus of quantum mechanics itself.
3.2.1. Overview of quantum mechanics
a. Relativistic theories. In the 1940s, Feynman, Schwinger, and Tomonaga each independently developed formulations of the theory of quantum electrodynamics (QED), which describes electromagnetic fields and electrons in a unified way. Where the mathematics of QED can be manipulated to yield precise predictions, its predictions (e.g., of the magnetic properties of free electrons and the spectrum of the hydrogen atom) have been confirmed to the last measurable detail. It correctly predicts that the electron -factor is rather than 2 as expected from previous theory, and it predicts the Lamb shift in the hydrogen spectrum (which changes an energy level by less than one part in ) to an accuracy of many decimal places. There is every reason to think that the theory would be equally precise in other areas, if it could be applied. In practice, owing to the difficulty of calculations, chemists and material scientists do not use QED. Its great contributions have been in high-energy physics and in its use as a prototype for other physical theories, such as quantum chromodynamics.
Earlier, in 1931, Dirac had developed a fully relativistic quantum mechanics which predicted electron spin (with a g-factor of 2) and provided a correct explanation for the splitting of certain spectral lines in hydrogen, reflecting shifts in energy levels by about one part in . Relativistic effects are large in the inner electron shells of heavy elements, but these shells are so tightly bound to the nucleus that they are chemically inert; most relativistic effects on valence electrons (especially those of light elements) are chemically negligible.* Since5
Dirac's theory is difficult to apply and describes effects that can often be neglected, it is not used in standard quantum mechanical calculations in chemistry and materials science.
b. Schrödinger's theory. Earlier still, in 1926, Schrödinger had developed a formulation of nonrelativistic quantum mechanics that remains the basis for calculations in quantum chemistry. Schrödinger described matter in terms of a wave equation (here shown with a scalar potential),
in which a system of particles with masses is described (neglecting spins) by the coordinate vector with components equaling , and a potential energy function . This is a partial differential equation in a -dimensional configuration space, and physically valid solutions, , are subject to a set of boundary, continuity, normalization, and particle-exchange symmetry conditions. Molecular structure calculations seek solutions describing bound, time-invariant systems, but this problem cannot be solved exactly even for a molecule as simple as . Nonetheless, because no simpler theory provides an acceptable description of the quantum mechanical behavior of molecular matter, the Schrödinger equation has become the basis of a host of approximation schemes. These approximations represent steps backward in fundamental physical theory, but steps forward in understanding real physical systems.
c. Schrödinger's theory applied to molecules. In order to understand the general nature of these approximation schemes, it is necessary to say a little more about Schrödinger's theory as applied to molecules.* In calculating the structure of isolated systems, the potential energy function is time independent, and is simply the total Coulomb energy for the interaction of each pair of charged particles
where is the distance between particles and and are their charges, and is the permittivity of free space. The potential energy function is thus based on a picture of particles with precise positions in ordinary space. The wave function determines the probability density function6
This expression is defined over the dimensional configuration space, hence it yields not only the particle density (and hence the charge density) at each point in 3-dimensional space, but also the probability density associated with each spatial configuration of particles. Owing to particle-exchange symmetry conditions and electrostatic repulsion, electron positions are not independent, but correlated. This precludes solving the dimensional problem as a set of coupled problems in three dimensions. (In the steady-state solutions of greatest chemical interest, this correlation of particle positions and motions is described by a wave function that is time independent, save for a time-varying phase factor.)
3.2.2. The Born-Oppenheimer PES
In chemistry and nanomechanical engineering, the full wave function gives more information than is necessary; indeed, the wave function per se is of little interest. Approximations and partial solutions can accordingly be of great value. Because each particle adds three dimensions and disproportionate computational cost, it is useful to partition problems into simpler subproblems when the resulting inaccuracies are not too severe.
Most computational techniques exploit the Born-Oppenheimer approximation, which treats the motion of electrons and nuclei separately. Even the lightest nucleus has times the inertia of an electron. The characteristic speeds and frequencies of nuclear motion are accordingly much lower than those of electronic motion. The Born-Oppenheimer approximation treats nuclei as motionless and computes the wave function and energy for a system of electrons in the presence of a fixed nuclear configuration. In this approximation, each nuclear configuration corresponds to a single electronic ground-state energy. This defines the ground-state Born-Oppenheimer potential energy function , where the vector specifies only nuclear coordinates and is quite unlike the simple Coulomb potential. In analyzing nuclear motions using this potential function, electronic motions are implicitly assumed to adjust without a time lag. The Born-Oppenheimer approximation breaks down when nuclear motions are fast enough (e.g., in high-energy collisions), when excited states occur at very low energies (this is rare in stable molecules), and when small changes in nuclear coordinates cause large changes in the electron wave function (for example in nearly degenerate states where the Jahn-Teller effect becomes important, as occurs in some symmetrical transition-metal complexes). Under ordinary conditions in which nuclear kinetic energies are less than electronic kinetic energies (and nuclear speeds are accordingly much smaller), the Born-Oppenheimer approximation usually gives an excellent account of molecular behavior.
In most nanomechanical systems, as in most chemical reactions, electron wave functions change smoothly with changes in molecular geometry. Under these conditions, there are no abrupt changes in electron distribution and energy, that is, no electronic transitions. In the absence of electronic transitions, and within the Born-Oppenheimer approximation, molecular dynamics depends only on the Born-Oppenheimer potential. If one knows this potential, or has an adequate approximation to it, then one can analyze molecular dynamics without reference to electronic behavior. (When electronic transitions occur, they place the system on another Born-Oppenheimer potential surface.) The
Born-Oppenheimer potential defines the potential energy surface, and can be used in both classical and quantum models of dynamics.
3.2.3. Molecular orbital methods
Practical calculations require further approximations. The most popular approaches result in a family of techniques known as molecular orbital methods; these make different approximations, yielding computations of widely varying accuracy and cost.
a. The independent-electron approximation. Molecular orbital methods begin with the independent electron approximation, in which each electron is treated as moving in the electrostatic potential that would result from the timeaverage distribution of the other electrons; this neglects the electrostatic electron correlation effects mentioned in Section 3.2.1c. Together with the BornOppenheimer approximation, this approximates the problem of solving a single Schrödinger equation in -dimensional configuration space as that of solving coupled Schrödinger equations in three-dimensional space, where is the number of electrons. The coupling is treated by the self-consistent field method: conceptually, each newly calculated electron wave function results in a new distribution of charge density, which changes the potential experienced by the other electrons, and so demands a new calculation; the resulting iterative process converges toward a set of single-electron wave functions, each consistent with the electrostatic potential of the rest (in practice, more efficient iterative procedures are employed). Each single-electron wave function corresponds to a molecular orbital having a particular energy and charge distribution. In this process, the overall many-electron wave function is described by a determinant based on these orbitals; this imposes particle-exchange symmetry conditions.
b. Approximate wave functions. In a typical quantum chemistry program, single-electron wave functions are approximated as weighted sums of simple (e.g., "Gaussian type") basis functions, and the programs vary the weighting of each basis function to form a wave function of minimum energy. Increasing the number of basis functions provides increased flexibility in shaping the wave function, enabling the construction of more accurate molecular orbitals; the number of basis functions must equal or (far better) exceed the number of electrons in the molecule. Computations on moderately large molecules are prohibitively expensive and time-consuming, however, because the cost of computing the energy of a single molecular geometry by these methods varies roughly as the fourth power of the number of basis functions.
c. Electron correlation and "levels of theory." Molecular orbital computations can be made more accurate by taking electron correlation into account using configuration interaction (CI) and related perturbation theory methods.* In CI methods, a "configuration" refers not to a spatial configuration of electrons (in the sense of a configuration space) but to a pattern of orbital occupancy: that7is, to a wave function formed from a particular set of single-electron wave functions. Judiciously mixing multiple wave functions, some representing excited states and each calculated using the independent-electron approximation, is equivalent to permitting correlated electron motions and hence reduces the errors of the independent-electron approximation. The set of possible configurations is combinatorially large; practical calculations often require selecting the most important configurations from a set of millions. The practical importance of electron correlation varies with the system and the phenomenon considered; it is the source of attractive forces between neutral, nonpolar molecules and is particularly important in calculating the energy of bonds far from equilibrium geometries (i.e., during the transition state of a chemical reaction). Configuration interaction calculations often converge slowly and expensively.
d. Semiempirical methods. The techniques just described (termed ab initio methods) make approximations in determining wave functions, but they use no parameters other than fundamental physical constants. Semiempirical molecular orbital methods (such as MNDO, AM1, and PM3) make further approximations in computing wave functions, but they compensate by introducing parameters for different atom types to fit results to experimental data. These methods neglect certain integrals that arise in more complete calculations and treat the compact, high-energy orbitals of inner-shell electrons as fixed. Semiempirical methods have a far lower computational cost, by a factor of 50 to 500 even for a molecule as small as propane (Clark, 1985), but vary in their accuracy in a complex manner (e.g., having special problems with hydrogen bonds, or with boron). There is considerable room for cleverness in semiempirical methods, which continue to improve (Stewart, 1990).
e. Results and accuracy. Molecular orbital methods can be used to calculate parameters such as charge distribution, polarizability, and ionization energies, but their chief application in the present context is to evaluate the energy as a function of molecular configuration: that is, the Born-Oppenheimer potential. This information suffices to determine equilibrium molecular geometries, corresponding to local or global energy minima, but finding them requires energy evaluations (often augmented by direct calculations of energy gradients) at many points in configuration space. More complex structures require evaluation at more points, hence computational cost scales with size even more adversely than the costs of single-point calculations would suggest.
The challenge of applying molecular orbital methods to chemical reactions is suggested by the great difference between the total molecular binding energy calculated by these methods (the energy required to separate a molecule into isolated electrons and nuclei) and the energy differences of chemical interest. The total binding energy for a small molecule is typically on the order of , and the energy of a single chemical bond is on the order of , but the energy required to rearrange bonds in a chemical reaction is typically , and the success of a solution-phase chemical process can be critically dependent on energy differences of . Thus, chemists often require that errors be less than or of the total energy, although larger errors in total energy can often be tolerated so long as energy differentials between similar structures are computed accurately enough.
This accuracy can be achieved using molecular orbital methods (for small structures), and such calculations can be used to design chemically-active groups for use in reactive devices in molecular manufacturing systems. Nonetheless, limitations associated with the expense, accuracy, and scalability of molecular orbital methods motivate the continuing popularity of empirically based models of molecules and chemical reactions.
3.3. Molecular mechanics
An alternative to performing quantum mechanical calculations of electronic structure is to approximate the Born-Oppenheimer potential directly in terms of the molecular geometry. Molecular mechanics methods (among others) take this approach, which is well suited to most problems of nanomechanical design and modeling.
3.3.1. The molecular mechanics approach
a. Overview. Organic chemists traditionally represent molecular structures with ball-and-stick models in which each ball represents an atom and each stick represents a bond. In quantum mechanical calculations, there is no a priori concept of a "chemical bond," and the behavior of bonding emerges afresh in each calculation. Molecular mechanics calculations, in contrast, begin with the concept of bonds, then use them as a basis for modeling the molecular potential energy surface. Molecular mechanics thus builds directly on a familiar and useful visualization of molecular structure.
Molecular mechanics programs use an approximate PES to calculate the properties of equilibrium molecular structures (that is, of local minima on the surface). This PES is developed by choosing functional forms and parameters that yield structures with properties that closely approximate the experimentally measured geometries, energies, and vibrational frequencies of molecules in the laboratory. Structural geometries are determined experimentally by (for example) -ray crystallography, microwave spectroscopy, and gas-phase electron diffraction; heats of formation by calorimetry; and vibrational frequencies by infrared spectroscopy. Molecular mechanics thus builds directly on a large body of experimental results regarding the mechanical properties of molecules.
Burkert and Allinger (1982) provide data on the typical accuracy of calculations performed on hydrocarbons using the MM2 model: estimated bond lengths typically match experimental values to within a few times , estimated bond angles typically match within about , or , and energies within a few times . These values are comparable in accuracy to the experimental data itself. Accuracies are lower among nonhydrocarbon structures, but are often good by nanomechanical standards (Section 4.4.3).
Burkert and Allinger compare the computational costs of molecular mechanics calculations to those of initio quantum mechanical calculations: for a small molecule (propylamine, with 13 atoms) cost favors molecular mechanics by a factor of . For larger molecules the difference becomes more pronounced: computational costs for molecular mechanics methods increase at between the second and third power of the number of atoms, rather than the higher powers characteristic of molecular orbital methods. As of 1992, molecular mechanics calculations on atom systems have become routine, and systems of to atoms are readily modeled using personal computers.
b. Limitations, content, and applications. Molecular mechanics systems have, however, been successfully applied to only a narrow range of molecular structures in configurations not too far from equilibrium. They use energy functions and parameters tailored to specific, local arrangements of atoms. Fortunately for nanomechanical engineering efforts, the most advanced molecular mechanics methods have been developed to model a class of structures that includes those most suitable for use as nanomechanical components-that is, structures built largely of carbon atoms (often augmented with one or more of the elements , and ) joined by strong, directional, covalent bonds. Structures of this class are the focus of organic chemistry; a subset of these structures comprises most of the molecular machinery of living systems.
Within this set, limitations remain. Aside from the small inaccuracies found in all structures, standard molecular mechanics programs cannot realistically describe certain structures. For example, they can model many stable structures, even when strained, but cannot model chemical transformations or systems on the verge of such transformations. Computational results must be examined with an eye for such invalidating conditions.
Molecular mechanics programs differ in their intended applications, with some designed for speed and others for accuracy, with some intended for broad classes of organic structures and others specialized for biomolecules. In general, systems intended for large biomolecules also place a premium on speed and give poor descriptions of structures with large strain energies; popular examples are AMBER (Weiner et al., 1984; Weiner et al., 1986) and CHARMM (Brooks et al., 1983). A widely used molecular mechanics model intended for a broad range of organic structures (including structures with large strain energies) is MM2 (Allinger, 1977), developed by Norman Allinger and his coworkers. Although in the process of being superseded by the similar, but more complex and accurate MM3 (Allinger et al., 1990; Allinger et al., 1989; Lii and Allinger, 1989a; Lii and Allinger, 1989b; Lii and Allinger, 1991), MM2 gives good results for a broad range of structures and is a standard model in the chemical literature. After evaluation and with some caveats, the MM2 model is used as an engineering approximation in much of the present work. Computations performed by MM2based software are used to characterize the minimum-energy configurations of structures, their stiffness, bearing properties, and the like; MM2 parameters are also used as a basis for several analytical models.
Molecular mechanics programs treat the total potential energy as a sum of terms accounting chiefly for bond stretching, bending, and torsion, and for van der Waals, overlap, and electrostatic interactions among nonbonded atoms. To develop a physical intuition for the mechanical behavior of molecular systems, one needs a feel for the nature and magnitude of these forces. Although physical intuition cannot substitute for a more precise analysis, it can be of great value in shaping designs and choosing what to analyze. The relationships and parameters that follow provide a good basis for physical understanding and also enable pocket-calculator estimates of the stiffness, force, and energy associated with molecular deformations. In the exploratory phase of design, one must test and
Table 3.1. Some atom types used in the MM2 program (Allinger, 1986) along with (rationalized) MM2 nonbonded interaction parameters. Note that the lone pair electrons associated with nitrogen and oxygen atoms are modeled by treating each pair as a pseudoatom. The full set of atom types described by MM2 is roughly twice as large as this list.
| MM2 code | Symbol | Type | | | Mass | Mass |
|---|---|---|---|---|---|---|
| 1 | 0.357 | 1.90 | 12.000 | 19.925 | ||
| 2 | alkene | 0.357 | 1.94 | 12.000 | 19.925 | |
| 3 | carbonyl | 0.357 | 1.94 | 12.000 | 19.925 | |
| 4 | acetylene | 0.357 | 1.94 | 12.000 | 19.925 | |
| 5 | hydrocarbon | 1.008 | 1.674 | |||
| 6 | 0.406 | 1.74 | 15.999 | 26.565 | ||
| 7 | carbonyl | 0.536 | 1.74 | 15.999 | 26.565 | |
| 8 | sp | 0.447 | 1.82 | 14.003 | 23.251 | |
| 11 | fluoride | 0.634 | 1.65 | 18.998 | 31.545 | |
| 12 | chloride | 1.950 | 2.03 | 34.969 | 58.064 | |
| 13 | bromide | 2.599 | 2.18 | 78.918 | 131.038 | |
| 14 | iodide | 3.444 | 2.32 | 126.900 | 210.709 | |
| 15 | sulfide | 1.641 | 2.11 | 31.972 | 53.087 | |
| 19 | silane | 1.137 | 2.25 | 27.977 | 46.454 | |
| 20 | lone pair | 0.130 | 1.20 | 0.000 | 0.000 | |
| 21 | alcohol | 0.292 | 1.20 | 1.008 | 1.674 | |
| 22 | cyclopropane | 0.357 | 1.90 | 12.000 | 19.925 | |
| 25 | phosphine | 1.365 | 2.18 | 30.994 | 51.464 |
See Section 3.3.2e for correction factors.
modify tentative concepts. This requires quick, accessible estimates of energies, forces, and stiffnesses, and of how they vary. For this, experience shows the value of graphs such as Figures 3.3 to 3.9.
3.3.2. The MM2 model
In molecular mechanics, bonds are defined by the types of the atoms they join, and these atom types can depend on both atomic number and bonding environment: for example, carbon atoms that participate only in single bonds are classified as a different type from those that participate in double or triple bonds. Table 3.1 lists some of the atom types used by the MM2 program and their nonbonded-interaction parameters (Section 3.3.2e).
a. Bond stretching. Bonds resist stretching and compression, tending toward an equilibrium length. The MM2 expression for the potential energy of bond stretching includes a cubic term to account for anharmonicity
In this expression, is the equilibrium bond length, is the actual bond length, is the stretching stiffness, and is the magnitude of a cubic correction, for which the MM2 model uses an invariant value of . (For ease of use,
Table 3.2. MM2 bond-stretching parameters (Allinger, 1986) for some common bond types; the full set is roughly six times larger.
| Bond type codes | | | Bond type |
|---|---|---|---|
| 460 | 1.113 | ||
| 440 | 1.523 | ||
| 960 | 1.337 | ||
| 1560 | 1.212 | ||
| 536 | 1.402 | ||
| 510 | 1.438 | ||
| 1080 | 1.208 | ||
| 510 | 1.392 | ||
| 323 | 1.795 | ||
| 230 | 1.949 | ||
| 220 | 2.149 | ||
| 610 | 0.600 | ||
| 560 | 1.381 | ||
| 460 | 0.600 | ||
| 460 | 0.942 | ||
| 781 | 1.470 | ||
| 297 | 1.880 | ||
| 291 | 1.856 | ||
| 321.3 | 1.815 | ||
| 185 | 2.332 | ||
| 440 | 1.501 | ||
| (cyclopropane) |
all MM2 expressions and parameters have been converted into SI units.) Table 3.2 lists some MM2 bond-stretching parameters.
b. Bond angle-bending. Two bonds to a shared atom define a bond angle (four bonds to a shared atom define six such angles). In the structures for which MM2 provides a good approximation, bond angles can be described as having preferred values with displacements countered by restoring forces; this describes amines poorly, and can describe trivalent but not pentavalent phosphorus (Burkert and Allinger, 1982). In its description of the potential energy of8
Table 3.3. MM2 bond angle-bending parameters (Allinger, 1986) for some common bond types; the full set is roughly ten times larger.
| Angle type (MM2 codes) | | | | Angle type |
|---|---|---|---|---|
| 0.450 | 109.47 | 1.911 | ||
| 0.360 | 109.39 | 1.909 | ||
| 0.320 | 109.40 | 1.909 | ||
| 0.650 | 109.50 | 1.911 |